Identifier
Values
[1] => [1] => [1] => 0
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 0
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 0
[2,1,3] => [2,1,3] => [2,1,3] => 0
[2,3,1] => [3,2,1] => [3,2,1] => 1
[3,1,2] => [3,1,2] => [3,1,2] => 0
[3,2,1] => [2,3,1] => [2,3,1] => 0
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 1
[1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 0
[1,4,3,2] => [1,3,4,2] => [1,3,4,2] => 0
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 2
[2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[2,4,3,1] => [3,2,4,1] => [3,2,4,1] => 1
[3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 0
[3,1,4,2] => [4,1,3,2] => [4,1,3,2] => 1
[3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 0
[3,2,4,1] => [2,4,3,1] => [2,4,3,1] => 1
[3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 2
[3,4,2,1] => [4,3,1,2] => [4,3,1,2] => 1
[4,1,2,3] => [4,1,2,3] => [4,1,2,3] => 0
[4,1,3,2] => [3,1,4,2] => [3,1,4,2] => 0
[4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 0
[4,2,3,1] => [2,3,4,1] => [2,3,4,1] => 0
[4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 1
[4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 0
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 1
[1,2,5,3,4] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[1,2,5,4,3] => [1,2,4,5,3] => [1,2,4,5,3] => 0
[1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,3,5,2,4] => [1,5,3,2,4] => [1,5,3,2,4] => 1
[1,3,5,4,2] => [1,4,3,5,2] => [1,4,3,5,2] => 1
[1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[1,4,2,5,3] => [1,5,2,4,3] => [1,5,2,4,3] => 1
[1,4,3,2,5] => [1,3,4,2,5] => [1,3,4,2,5] => 0
[1,4,3,5,2] => [1,3,5,4,2] => [1,3,5,4,2] => 1
[1,4,5,2,3] => [1,5,4,3,2] => [1,5,4,3,2] => 2
[1,4,5,3,2] => [1,5,4,2,3] => [1,5,4,2,3] => 1
[1,5,2,3,4] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[1,5,2,4,3] => [1,4,2,5,3] => [1,4,2,5,3] => 0
[1,5,3,2,4] => [1,3,5,2,4] => [1,3,5,2,4] => 0
[1,5,3,4,2] => [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,5,4,2,3] => [1,4,5,3,2] => [1,4,5,3,2] => 1
[1,5,4,3,2] => [1,4,5,2,3] => [1,4,5,2,3] => 0
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Description
The nesting alignments of a signed permutation.
A nesting alignment of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1\leq i, j \leq n$ such that
  • $-i < -j < -\pi(j) < -\pi(i)$, or
  • $-i < j \leq \pi(j) < -\pi(i)$, or
  • $i < j \leq \pi(j) < \pi(i)$.
Map
to signed permutation
Description
The signed permutation with all signs positive.
Map
Corteel
Description
Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.